Publications
For a detailed list of publications, please see here (PDF). Most of my papers (including citations) can also be found on INSPIRE.
Preprints

algebras of classical field theories and the Batalin—Vilkovisky formalism
with Branislav Jurco, Lorenzo Raspollini, and Christian Sämann
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We review in detail the Batalin—Vilkovisky formalism and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an algebra and how quasiisomorphisms between algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern—Simons theories and give some useful shortcuts in usually rather involved computations.
Journal Papers

Supersymmetric Yang—Mills theory as higher ChernSimons theory
JHEP 1707 (2017) 111
with Christian Sämann
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We observe that the string field theory actions for the topological sigma models describe higher or categorified ChernSimons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher ChernSimons theory on the ambitwistor space of fourdimensional spacetime. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang—Mills theory.

Higher groupoid bundles, higher spaces, and selfdual tensor field equations
Fortschr Phys 64 (2016) 674
with Branislav Jurco and Christian Sämann
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We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a selfcontained review on simplicial sets as models of categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain sixdimensional superconformal field theories via a PenroseWard transform of higher groupoid bundles over a twistor space. This construction reduces the search for nonAbelian selfdual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.

Tduality of GreenSchwarz superstrings on
JHEP 1512 (2015) 104
with Michael Abbott, Jeff Murugan, Silvia Penati, Antonio Pittelli, Dmitri Sorokin, Per Sundin, Justine Tarrant, and Linus Wulff
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We verify the selfduality of GreenSchwarz supercoset sigma models on backgrounds () under combined bosonic and fermionic Tdualities without gauge fixing kappa symmetry. We also prove this property for superstrings on () described by supercoset sigma models with the isometries governed by the exceptional Lie supergroups () and (), which requires an additional Tdualisation along one of the spheres. Then, by taking into account the contribution of nonsupercoset fermionic modes (up to the second order), we provide evidence for the Tselfduality of the complete type IIA and IIB GreenSchwarz superstring theory on () backgrounds with RamondRamond fluxes. Finally, applying the Buscherlike rules to Tdualising supergravity fields, we prove the Tselfduality of the whole class of the superbackgrounds with RamondRamond fluxes in the context of supergravity.

Semistrict higher gauge theory
JHEP 1504 (2015) 087
with Branislav Jurco and Christian Sämann
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We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2bundles with connective structures. Principal 2bundles are obtained in terms of weak 2functors from the Cech groupoid to weak Lie 2groups. As is demonstrated, some of these Lie 2groups can be differentiated to semistrict Lie 2algebras by a method due to Severa. We further derive the full description of connective structures on semistrict principal 2bundles including the nonlinear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a nonAbelian tensor multiplet taking values in a semistrict Lie 2algebra.

Secret symmetries of type IIB superstring theory on
J Phys A Math Theor 47 (2014) 455402
with Antonio Pittelli and Alessandro Torrielli
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We establish features of socalled Yangian secret symmetries for type IIB superstring backgrounds thus verifying the persistence of such symmetries to this new instance of the AdS/CFT correspondence. Specifically, we find two different secret symmetry generators which appear unrelated. One generator, anticipated from the previous literature, is more naturally embedded in the algebra governing the integrable scattering problem. The other generator is new and more elusive, and somewhat closer in its form to its higherdimensional counterpart. All of these symmetries respect leftright crossing. In addition, by considering the interplay between left and right representations, we gain a new perspective on the case. We also study the realisation of the Yangian in backgrounds thus establishing a new incarnation of the Beisertde Leeuw construction.

Sixdimensional superconformal field theories from principal 3bundles over twistor space
Lett Math Phys 104 (2014) 1147
with Christian Sämann
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We construct manifestly superconformal field theories in six dimensions which contain a nonAbelian tensor multiplet. In particular, we show how principal 3bundles over a suitable twistor space encode solutions to these selfdual tensor field theories via a PenroseWard transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2bundles in that the socalled Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3bundles such as the relevant gauge transformations. We thus arrive at the nonAbelian differential cohomology that describes principal 3bundles with connective structure.

NonAbelian tensor multiplet equations from twistor space
Commun Math Phys 328 (2014) 527
with Christian Sämann
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We establish a PenroseWard transform yielding a bijection between holomorphic principal 2bundles over a twistor space and nonAbelian selfdual tensor fields on sixdimensional flat spacetime. Extending the twistor space to supertwistor space, we derive sets of manifestly and supersymmetric nonAbelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for nonAbelian supersymmetric selfdual strings.

On twistors and conformal field theories from six dimensions
J Math Phys 54 (2013) 013507
with Christian Sämann
To the memory of Francis Dolan
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We discuss chiral zerorestmass field equations on sixdimensional spacetime from a twistorial point of view. Specifically, we present a detailed cohomological analysis, develop both Penrose and Penrose—Ward transforms, and analyse the corresponding contour integral formulae. We also give twistor space action principles. We then dimensionally reduce the twistor space of sixdimensional spacetime to obtain twistor formulations of various theories in lower dimensions. Besides wellknown twistor spaces, we also find a novel twistor space amongst these reductions, which turns out to be suitable for a twistorial description of selfdual strings. For these reduced twistor spaces, we explain the Penrose and Penrose—Ward transforms as well as contour integral formulae.

Contact manifolds, contact instantons, and twistor geometry
JHEP 1207 (2012) 074
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Recently, Källen & Zabzine computed the partition function of a twisted supersymmetric Yang—Mills theory on the fivedimensional sphere using localisation techniques. Key to their construction is a fivedimensional generalisation of the instanton equation to which they refer as the contact instanton equation. Subject of this article is the twistor construction of this equation when formulated on Kcontact manifolds and the discussion of its integrability properties. We also present certain extensions to higher dimensions and supersymmetric generalisations.

A twistor description of sixdimensional super Yang—Mills theory
JHEP 1205 (2012) 020
with Christian Sämann and Robert Wimmer
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We present a twistor space that describes super nulllines on sixdimensional superspace. We then show that there is a onetoone correspondence between holomorphic vector bundles over this twistor space and solutions to the field equations of super Yang—Mills theory. Our constructions naturally reduce to those of the twistorial description of maximally supersymmetric Yang—Mills theory in four dimensions.

Shaping up BPS states with matrix model saddle points
J Phys A Math Theor 43 (2010) 465402
with Diego H Correa
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We provide analytical results for the probability distribution of a family of wavefunctions of a quantum mechanics model of commuting matrices in the largeN limit. These wavefunctions describe the strong coupling limit of BPS states of supersymmetric Yang—Mills theory. Therefore, in the large limit, they should be dual to classical solutions of type IIB supergravity that asymptotically approach . Each probability distribution can be described as the partition function of a matrix model (different wavefunctions correspond to different matrix model potentials) which we study by means of a saddle point approximation. These saddle point solutions are given in terms of (fivedimensional) hypersurfaces supporting density distributions of eigenvalues.

A first course on twistors, integrability and gluon scattering amplitudes
J Phys A Math Theor 43 (2010) 393001
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These notes accompany an introductory lecture course on the twistor approach to supersymmetric gauge theories aimed at earlystage PhD students. It was held by the author at the University of Cambridge during the Michaelmas term in 2009. The lectures assume a working knowledge of differential geometry and quantum field theory. No prior knowledge of twistor theory is required.

Marginal deformations and 3algebra structures
Nucl Phys B 826 (2010) 456
with Nikolas Akerblom and Christian Sämann
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We study marginal deformations of superconformal Chern—Simons matter theories that are based on 3algebras. For this, we introduce the notion of an associated 3product, which captures very general gauge invariant deformations of the superpotentials of the BLG and ABJM models. We then use supergraph techniques to compute the twoloop beta functions of these and multitrace deformations. Besides confirming conformal invariance of both the BLG and ABJM models, we also verify that the recently proposed betadeformations of the models are indeed marginal to the order we are considering.

A connection between twistors and superstring sigma models on coset superspaces
JHEP 0909 (2009) 071
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We consider superstring sigma models that are based on coset superspaces in which arises as the fixed point set of an order automorphism of . We show by means of twistor theory that the corresponding firstorder system, consisting of the Maurer—Cartan equations and the equations of motion, arises from a dimensional reduction of some generalised selfdual Yang—Mills equations in eight dimensions. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations including their hidden symmetry structures and thus on the properties of their gauge theory duals.

Twistor actions for selfdual supergravities
Commun Math Phys 288 (2009) 97
with Lionel J Mason
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We give holomorphic Chern—Simonslike action functionals on supertwistor space for selfdual supergravity theories in four dimensions, dealing with supersymmetries, the cases where different parts of the symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable operator on a supertwistor space, i.e., on regions in . For , we also give a formulation that does not require the choice of a background.

Dual superconformal symmetry from superstring integrability
Phys Rev D 78 (2008) 126004
with Niklas Beisert, Riccardo Ricci, and Arkady A Tseytlin
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We discuss duality transformations in the classical superstring and their effect on the integrable structure. Tduality along four directions in Poincare parametrization of maps the bosonic part of the superstring action into itself. On bosonic level, this duality may be understood as a symmetry of the firstorder (phase space) system of equations for the coset components of the current. The associated Lax connection is invariant modulo the action of an automorphism. We then show that this symmetry extends to the full superstring, provided one supplements the transformation of the bosonic components of the current with a transformation on the fermionic ones. At the level of the action, this symmetry can be seen by combining the bosonic duality transformation with a similar one applied to part of the fermionic superstring coordinates. As a result, the full superstring action is mapped into itself, albeit in a different kappasymmetry gauge. One implication is that the dual model has the same superconformal symmetry group as the original one, and this may be seen as a consequence of the integrability of the superstring. The invariance of the Lax connection under the duality implies a map on the full set of conserved charges that should interchange some of the Noether (local) charges with hidden (nonlocal) ones and vice versa.

On Tduality and integrability for strings on AdS backgrounds
JHEP 0712 (2007) 082
with Riccardo Ricci and Arkady A Tseytlin
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We discuss an interplay between Tduality and integrability for certain classical nonlinear sigma models. In particular, we consider strings on the background and perform Tduality along the four isometry directions of in the Poincare patch. The Tdual of the sigma model is again a sigma model on an space. This classical Tduality relation was used in the recently uncovered connection between lightlike Wilson loops and MHV gluon scattering amplitudes in the strong coupling limit of the AdS/CFT duality. We show that the explicit coordinate dependence along the Tduality directions of the associated Lax connection (flat current) can be eliminated by means of a field dependent gauge transformation. As a result, the gauge equivalent Lax connection can easily be Tdualized, i.e. written in terms of the dual set of isometric coordinates. The Tdual Lax connection can be used for the derivation of infinitely many conserved charges in the Tdual model. Our construction implies that local (Noether) charges of the original model are mapped to nonlocal charges of the Tdual model and vice versa.

Selfdual supergravity and twistor theory
Class Quant Grav 24 (2007) 6287
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By generalizing and extending some of the earlier results derived by Manin and Merkulov, a twistor description is given of fourdimensional extended (gauged) selfdual supergravity with and without cosmological constant. Starting from the category of dimensional complex superconformal supermanifolds, the categories of dimensional complex quaternionic, quaternionic Kähler and hyperKähler rightchiral supermanifolds are introduced and discussed. We then present a detailed twistor description of these types of supermanifolds. In particular, we construct supertwistor spaces associated with complex quaternionic rightchiral supermanifolds, and explain what additional supertwistor data allows for giving those supermanifolds a hyperKähler structure. In this way, we obtain a supersymmetric generalization of Penrose's nonlinear graviton construction. We furthermore give an alternative formulation in terms of a supersymmetric extension of LeBrun's Einstein bundle. This allows us to include the cases with nonvanishing cosmological constant. We also discuss the bundle of local supertwistors and address certain implications thereof. Finally, we comment on a real version of the theory related to Euclidean signature.

Hidden symmetries and integrable hierarchy of the supersymmetric Yang—Mills equations
Commun Math Phys 275 (2007) 685
with Alexander D Popov
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We describe an infinitedimensional algebra of hidden symmetries of supersymmetric Yang—Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries recursively generated from supertranslations. Presumably, the existence of such nonlocal symmetries underlies the observed integrable structures in quantum SYM theory.

The topological B model on a minisupertwistor space and supersymmetric Bogomolny monopole equations
JHEP 0510 (2005) 058
with Alexander D Popov and Christian Sämann
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In the recent paper
hepth/0502076, it was argued that the open topological B model whose target space is a complex dimensional minisupertwistor space with  and branes added corresponds to a super Yang—Mills theory in three dimensions. Without the branes, this topological B model is equivalent to a dimensionally reduced holomorphic Chern—Simons theory. Identifying the latter with a holomorphic BFtype theory, we describe a twistor correspondence between this theory and a supersymmetric Bogomolny model on . The connecting link in this correspondence is a partially holomorphic Chern—Simons theory on a Cauchy—Riemann supermanifold which is a real onedimensional fibration over the minisupertwistor space. Along the way of proving this twistor correspondence, we review the necessary basic geometric notions and construct action functionals for the involved theories. Furthermore, we discuss the geometric aspect of a recently proposed deformation of the minisupertwistor space, which gives rise to mass terms in the supersymmetric Bogomolny equations. Eventually, we present solution generating techniques based on the developed twistorial description together with some examples and comment briefly on a twistor correspondence for super Yang—Mills theory in three dimensions.

Nontrivial solutions of the Seiberg—Witten equations on the noncommutative fourdimensional Euclidean space
Proc Steklov Inst 251 (2005) 127
with Alexander D Popov and Armen G Sergeev
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Noncommutative Seiberg—Witten equations on the noncommutative Euclidean space are studied that are obtained from the standard Seiberg—Witten equations on by replacing the usual product with the deformed Moyal starproduct. Nontrivial solutions of these noncommutative Seiberg—Witten equations are constructed that do not reduce to solutions of the standard Seiberg—Witten equations on for . Such solutions of the noncommutative equations on exist even when the corresponding commutative Seiberg—Witten equations on do not have any nontrivial solutions.

On hidden symmetries of a super gauge theory and twistor string theory
JHEP 0502 (2005) 018
To the memory of Gerhard Soff
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We discuss infinitedimensional hidden symmetry algebras (and hence an infinite number of conserved nonlocal charges) of the extented selfdual super Yang—Mills equations for general by using the supertwistor correspondence. Furthermore, by enhancing the supertwistor space, we construct the extended selfdual super Yang—Mills hierarchies, which describe infinite sets of graded Abelian symmetries. We also show that the open topological B model with the enhanced supertwistor space as target manifold will describe the hierarchies. Furthermore, these hierarchies will in turn, by a supersymmetric extension of Ward's conjecture, reduce to the super hierarchies of integrable models in dimensions.

Topological B model on weighted projective spaces and selfdual models in four dimensions
JHEP 0409 (2004) 007
with Alexander D Popov
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It was recently shown by Witten on the basis of several examples that the topological B model whose target space is a CalabiYau (CY) supermanifold is equivalent to holomorphic Chern—Simons (hCS) theory on the same supermanifold. Moreover, for the supertwistor space as target space, it has been demonstrated that hCS theory on is equivalent to selfdual super Yang—Mills (SYM) theory in four dimensions. We consider as target spaces for the B model the weighted projective spaces with two fermionic coordinates of weight and , respectively  which are CY supermanifolds for  and discuss hCS theory on them. By using twistor techniques, we obtain certain field theories in four dimensions which are equivalent to hCS theory. These theories turn out to be selfdual truncations of SYM theory or of its twisted (topological) version.

Constraint and super Yang—Mills equations on the deformed superspace
JHEP 0403 (2004) 048
with Christian Sämann
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It has been known for quite some time that the super Yang—Mills equations defined on fourdimensional Euclidean space are equivalent to certain constraint equations on the Euclidean superspace . In this paper we consider the constraint equations on a deformed superspace a la Seiberg and derive the deformed super Yang—Mills equations. In showing this, we propose a super Seiberg—Witten map.

Seiberg—Witten monopole equations on noncommutative
J Math Phys 44 (2003) 4527
with Alexander D Popov and Armen G Sergeev
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It is well known that, due to vanishing theorems, there are no nontrivial finite action solutions to the Abelian SeibergWitten (SW) monopole equations on Euclidean fourdimensional space . We show that this is no longer true for the noncommutative version of these equations, i.e., on a noncommutative deformation of there exist smooth solutions to the SW equations having nonzero topological charge. We introduce action functionals for the noncommutative SW equations and construct explicit regular solutions. All our solutions have finite energy. We also suggest a possible interpretation of the obtained solutions as codimension four vortexlike solitons representing  and antibranes in a anti brane system in type II superstring theory.

Noncommutative instantons via dressing and splitting approaches
JHEP 0212 (2002) 060
with Zalan Horvath and Olaf Lechtenfeld
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Almost all known instanton solutions in noncommutative Yang—Mills theory have been obtained in the modified ADHM scheme. In this paper we employ two alternative methods for the construction of the selfdual BPST instanton on a noncommutative Euclidean fourdimensional space with selfdual noncommutativity tensor. Firstly, we use the method of dressing transformations, an iterative procedure for generating solutions from a given seed solution, and thereby generalize Belavin's and Zakharov's work to the noncommutative setup. Secondly, we relate the dressing approach with Ward's splitting method based on the twistor construction and rederive the solution in this context. It seems feasible to produce nonsingular noncommutative multiinstantons with these techniques.

Solitonantisoliton scattering configurations in a noncommutative sigma model in dimensions
JHEP 0206 (2002) 055
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In this paper we study the noncommutative extension of a modified sigma model in dimensions. Using the method of dressing transformations, an iterative approach for the construction of solutions from a given seed solution, we demonstrate the construction of multisoliton and solitonantisoliton configurations for general . As illustrative examples we discuss solitons and consider the headon collision of a soliton and an antisoliton explicitly, which will result in a angle scattering. Further we discuss the headon collision of one soliton with two antisolitons. This results in a angle scattering.

Artificiality of multifractal phase transitions
Phys Lett A 266 (2000) 276
with Jürgen Schmiegel and Martin Greiner
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A multifractal phase transition is associated to a nonanalyticity in the generalised dimensions. We show that its occurrence is an artifact of the asymptotic scaling behaviour of integral moments and that it is not observed in an analysis based on differential point correlation densities.
Proceedings

Twistors and aspects of integrability of selfdual SYM theory
Proc Intern Workshop on Supersymmetries and Quantum Symmetries 1 (2005) 44 (Eds E Ivanov and B Zupnik)
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With the help of the PenroseWard transform, which relates certain holomorphic vector bundles over the supertwistor space to the equations of motion of selfdual SYM theory in four dimensions, we construct hidden infinitedimensional symmetries of the theory. We also present a new and shorter proof (cf.
hepth/0412163) of the relation between certain deformation algebras and hidden symmetry algebras. This article is based on a talk given by the author at the Workshop on Supersymmetries and Quantum Symmetries 2005 at the BLTP in Dubna, Russia.

Twistor geometry and gauge theory
Proc Modave Summer School Math Phys 1 (2005) 248 (Eds V Bouchart and A Wijns)
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These are lecture notes supplementing a threehour introductory course on twistor geometry and gauge theory given at the Modave Summer School on Mathematical Physics in June 2005. In the first lecture, we discuss the basics of twistor geometry and a supersymmetric extension thereof. The second lecture introduces the Penrose transform. Finally, the third lecture deals with the supertwistor correspondence relating a certain theory on supertwistor space to super gauge theory in four dimensions. Of course, I have tried to track down and remove all mistakes from these notes, but nevertheless it is rather unlikely I succeeded in doing so. In case you find any, please let me know.
Other Contributions

On supertwistor geometry and integrability in super gauge theory
PhD thesis (2006)
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In this thesis, we report on different aspects of integrability in supersymmetric gauge theories. The main tool of investigation is twistor geometry. In trying to be selfcontained, we first present a brief review about the basics of twistor geometry. We then focus on the twistor description of various gauge theories in four and three spacetime dimensions. These include selfdual supersymmetric Yang—Mills (SYM) theories and relatives, nonselfdual SYM theories and supersymmetric Bogomolny models. Furthermore, we present a detailed investigation of integrability of selfdual SYM theories. In particular, the twistor construction of infinitedimensional algebras of hidden symmetries is given and exemplified by deriving affine extensions of internal and spacetime symmetries. In addition, we derive selfdual SYM hierarchies within the twistor framework. These hierarchies describe an infinite number of flows on the respective solution space, where the lowest level flows are spacetime translations. We also derive infinitely many nonlocal conservation laws.

On topologically nontrivial field configurations in noncommutative field theories
Diploma thesis (2003)
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In this diploma thesis we derive nonperturbative solutions to the equations of motion of various noncommutative field theoretic models. Firstly, we consider pure Yang—Mills theory defined on noncommutative Euclidean fourdimensional space and derive the noncommutative version of the selfdual Belavin—Polyakov—Schwarz—Tyupkin instanton solution using the method of dressing transformations, an iterative procedure for generating solutions from a given seed solution. In doing so, we generalize Belavin's and Zakharov's work to the noncommutative setup. Furthermore, we relate the dressing approach with Ward's splitting method based on the twistor construction and rederive the solution in this context. Secondly, we study the noncommutative extension of a modified sigma model in dimensions. Using the method of dressing transformations, we demonstrate the construction of multisoliton and solitonantisoliton configurations for general . As examples, we investigate solitons and solitonantisoliton scattering configurations. Thirdly, we consider the noncommutative version of the Abelian Seiberg—Witten monopole equations on Euclidean fourdimensional space. It has been known for quite some time that due to vanishing theorems there are no regular square integrable solutions to these equations on Euclidean spaces. For a vanishing spinor field, these equations reduce to the Abelian antiselfdual Yang—Mills equations, which have no regular solutions either. However, Nekrasov and Schwarz have shown that regular Abelian instanton solutions do exist on noncommutative Euclidean spaces. We generalize their results by constructing several regular solutions to the noncommutative Abelian Seiberg—Witten monopole equations.
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